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PREPARATORY TEST 2011
COMPLETE SOLUTIONS
EULER (7th) – LAGRANGE (8th) – NEWTON (9th)
1.
The sum of the prime factors of 20 (2 x 2 x 5) is 9.
2.
The value of Y is 150°, that of X is (180° - 150°) 30°. The value of 2Y + X is 330°.
3.
April 14 2011 (2011 is not a leap year) will come 365 days (52 x 7 + 1) after April 14 2010. This day
will come 52 weeks and 1 day after April 14 2010. It will fall on a Thursday.
4.
The result of 0.1 x 0.1 x 0.1 x 10 is 0.01, which is 1%.
5.
The exception is 12 (1, 2, 3, 4, 6, 12) which has 6 factors.
6.
A number divided by 3 leaves a remainder of 1. This number is of the form 3N + 1. Replacing N
by 0, 1, 2, 3, … we get the following set of number {1, 4, 7, 10, 13, 16, 19, 22, …}. Another number
divided by 3 leaves a remainder of 2. This number is of the form 3N + 2. Replacing N by 0, 1, 2, 3,
… we get the following set of numbers {2, 5, 8, 11, 14,17, 20, 23, …}. The product of these 2
numbers could not be 25 because this number can only be obtained by the product of 5 x 5 or
1 x 25.
7.
5 1/3 x 2 1/4 x 3/8 x 4/9 = 16/3 x 9/4 x 3/8 x 4/9 = 16/8 = 2.
7.
No element of the set verifies x2 < -x. Actually (-1)2 = 1,
22 > -2, (3/4)2 > -3/4, (-4)2 > 4 and (1/3)2 > -1/3.
9.
It takes 9 circles to completely surround the 3
tangent circles.
10. The 4-digit perfect square 2mn1 could be
(512) 2601 or (492) 2401. The possible
value of m + n is 6 or 4.
11. The sum of 2 integers is 1 and their product
Is -2. The factors of 2 are 1 and 2. We can
get a product of -2 in two different ways:
1 x -2 and -1 x 2. The sum of the two numbers
being 1, we can conclude that the
two numbers are 2 and -1. Their quotient
could be -2 or -1/2.
12. The number of seconds in one hour is (1h = 60min = 60 x 60s) 3 600 seconds. 3 600
minutes are equal to 60 hours (60h = 60 x 60min = 3 600min).
13. The values of y lie between 0 and 3. Therefore y is always positive. The values of x lie between -4
and 0. Therefore x is always negative. The answer y - x >0 (y > x) is always true.
14. The average of -1/2 and 1/3 (-1/2 + 1/3 = -3/6 + 2/6 = -1/6 and -1/6 ÷ 2 = -1/12) is -1/12.
15. The distance that the car covers when each tire makes 100 revolutions is (100 x π x 1) 100π, which
is approximately 314 m.
16. There are 5 balls left in the triangle. Mathilda cannot win the game. Actually, if she removes 1 ball,
Mathew removes 3 balls (taking the 14th ball) and wins the game. If Mathilda takes out 2 balls,
Mathew takes out 2 balls and wins again. If Mathilda removes 3 balls from the triangle, Mathew
wins again by removing 1 ball. Mathew was certain to win because he removed the 10th ball from
the triangle. The person who removes the 10th ball is certain to remove the 14th ball. The person
who removes the 6th ball is certain to remove the 10th. Finally, the person who takes out the 2nd ball
is sure to remove the 6th one. Applying this method, the player who plays first is in a position to win
any game.
17. The result of 1 + 3 = 4 = 22. The result of 1 + 3 + 5 = 9
= 32. The result of 1 + 3 + 5 + 7 = 42 = 16. The sum
of the series of the first ‘’n’’ consecutive odd numbers
is always n2. To know how many terms there are in a
series of consecutive odd numbers just calculate the
half sum of the first and last terms of the series. The
series 1 + 3 + 5 + 7 … + 41 has ((1 + 41) ÷ 2)
21 terms. The result of this series is 212 (441).
18. Tooth F is meshed with teeth 3 and 4. When the gear
on the right turns in the direction of the arrow, the lettered
teeth turn in the following ordered sequence: F, G, H, A, B,
C….The numbered teeth of the left gear turn in the following ordered sequence: 3-4, 2-3, 1-2, 9-1,
8-9, …. The sequence of the meshed teeth will be 3F4, 2G3, 1H2, 9A1 … . The teeth that will be
meshed when tooth A falls into the position now occupied by tooth F are 1A9.
19. Four neighbours have bought a snow blower and have shared the cost equally. If x is the price of
the snow blower, each neighbour has paid x/4. If there were only three neighbours, each one would
have paid x/3. We can write that x/3 - x/4 = 105, 4x/12 - 3x/12 = 105, 4x - 3x = 1 260 and
x = $1 260. If 5 neighbours had bought the blower, each one would have paid (1 260 ÷ 5) $252.
20. The radius of the circle is 2. The length of the circle is (2π2) 4π. The length of a quarter-circle is π.
21. It’s average speed is (150 km ÷ 15min) 10 km/min or 600 km/h.
22. Mathew has received 1 + 3 + 5 + … = 1 000 000. We
know that n2 = 1 000 000 and that n = 1 000. He has
received a certain amount (1, 3, 5, …) every week for
1 000 weeks. The amount x received on the 1 000th
week is given by the equation: (1 + x) ÷ 2 = 1 000 (see
number 17). This equation becomes 1 + x = 2 000.
We find that x = 1 999. The last week, Mathew
received $1 999.
23. The value of X is given by equation 180° - X + 180° - X + 80° = 180°. We find that X = 130°.
24. The value of 225 is 25 x 25 x 25 x 25 x 25 = 32 x 32 x 32 x 32 x 32 = 33 554 432 = 3.4 x 107. The
closest answer is 107.
25. Pythagoras’ theorem states that, in a right triangle, if
c is the length of the hypotenuse and a and b are the
the lengths of the other two sides, then a2 + b2 = c2. If
a and b are two sides of a square (a = b = 1) and c
is the length of a diagonal, then c2 = 12 + 12 = 2 and
c = √2.
26. Container A holds 3 litres of water. Recipient B
holds a homogeneous mixture of 2 litres of wine
and 1 litre of water. A litre of mixture B is poured in
container A. This litre contains 2/3 of a litre of wine and 1/3 of a litre of water. Recipient A now
contains 3 1/3 litres of water and 2/3 of a litre of wine. Recipient A contains a total of (3 1/3 + 2/3) 4
litres of a mixture of water and wine. Wine represents (2/3 ÷ 4 = 2/12) 1/6 of the mixture in A.
27. Because I get a remainder of 1 each time I divide the number by 2, 3, 4, or 5, this number must be
of the form 60N + 1. If the number was divisible by 2, 3, 4, and 5 (2 x 3 x 2 x 5), it would be a
multiple of 60 (it would be of the form 60N). However, there is a remainder of 1, so it must be of the
form 60N + 1. Replacing N by 0, 1, 2, 3, 4, … we get the numbers 1, 61, 121, 181, 241, … . The
number we are looking for is 181 and the sum of its digits is (1 + 8 + 1) 10.
28. The graph of line 3y - x = 6 is shown in the
diagram. The number of grid points on this
line is given by the equation y = x/3 + 2. This
equation tells us that y is an integer only when
x is a multiple of 3. For the interval -6 ≤ x ≤ 4,
y is an integer for x = {-6, -3, 0, 3}. There are
4 grid points on this line for -6 ≤ x ≤ 4.
29. The volume of cylinder A is (π42 x 20)
320 π cm3. The volume of B is given by the
equation π102 x h = 320π. This equation
becomes 100h = 320, which gives h= 3.2 cm.
30. The edges of the cube shown in the diagram
have a value of 1. Points B, C, D, and E are
4 vertices of the cube, whereas point A is a
point of one of its edges. Triangle BDC is an
isosceles right triangle (angle D = 90° and
DA = DC = 1). The length of line segment
BC is given by BC2 = 12 + 12. The value
of segment BC is √2. Now look at triangle
BDC in which we have drawn the height h
of side BC. Again, we can write that
h2 + (√2/2)2 = 12. We find that h = √2/2.
By the way, triangles ABC and EBC are
also isosceles triangles. Triangle EBC
is in reality an equilateral triangle
because its sides are all equal to √2.
31. Given that all the pizzas have the same thickness and are covered by the same toppings, the best
buy is given by the cost per cm2 of pizza. Member D has made the best buy because he paid his
pizza a mere 2.31¢ per cm2 (E made the second best buy because he paid only 2.38¢/cm2.)